Generalized solutions to linearized equations of Thermo-elastic solid and viscous thermo-fluid

Within the framework of continuum mechanics, the full description Of joint motion of elastic bodies and compressible viscous fluids with taking into account thermal effects is given by the system consisting of the mass, momentum, and energy balance equations, the first and the second laws of thermodynamics, and an additional set of thermo-mechanical state laws. The present paper is devoted to the investigation of this system. Assuming that variations of the physical characteristics of the thermo-mechanical system of the fluid and the solid are small about some rest state, we derive the linearized non-stationary dynamical model, prove its well-posedness, establish additional refined global integral bounds for solutions, and further deduce the linearized incompressible models and models incorporating absolutely rigid skeleton, as asymptotic limits.Comment: submitted to EJD

Mathematical models of seismics in composite media: elastic and poro-elastic components

In the present paper we consider elastic and poroelastic media having a common interface. We derive the macroscopic mathematical models for seismic wave propagation through these two different media as a homogenization of the exact mathematical model at the microscopic leve

A compactness lemma of aubin type and its application to degenerate parabolic equations

Let Ω ⊂ Rⁿ be a regular domain and Φ(s) ∈ C loc (R) be a given function. If M⊂ L₂ (0, T ; W½ (Ω)) ∩ L ∞ (Ω × (0, T )) is bounded and the set {∂t Φ(v)|v ∈ M} is bounded in L₂ (0, T ; W-¹₂ (Ω)), then there is a sequence {vk} ∈ M such that vk ⇀ v ∈ L₂ (0,T ; W¹₂ (Ω)), and vk → v, Φ(vk) → Φ(v) a.e. in Ωτ = Ω × (0, T). This assertion is applied to prove solvability of the one-dimensional initial and boundary-value problem for a degenerate parabolic equation arising in the Buckley-Leverett model of two-phase filtration. We prove existence and uniqueness of a weak solution, establish the property of finite speed of propagation and construct a self-similar solutio

Seismic in composite media: elastic and poroelastic components

In the present paper we consider elastic and poroelastic media having a common interface. We derive the macroscopic mathematical models for seismic wave propagation through these two different media as a homogenization of the exact mathematical model at the microscopic level. They consist of seismic equations for the each component and boundary conditions at the common interface, which separates different medi

The Muskat problem at the microscopic level for a single capillary

We consider the evolution of the free boundary separating two immiscible viscous fluids with different constant densities. The motion is described by the Stokes equations driven by the input pressure and gravity force. For flows in a bounded domain Ω⊂R², we prove existence and uniqueness of classical solutions and make an emphasis on the study of properties of the moving boundary separating the two fluidsyesBelgorod State National Research Universit

Solvability of a free-boundary problem describing the traffic flows

We study a mathematical model of the vehicle traffc on straight freeways, which describes the traffc flow by means of equations of one dimensional motion of the isobaric viscous gas. The corresponding free boundary problem is studied by means of introduction of Lagrangian coordinates, which render the free boundary stationary. It is proved that the equivalent problem posed in a time-independent domain admits unique local and global in time classical solutions. The proof of the local in time existence is performed with stan- dard methods, to prove the global in time existence the system is reduced to a system of two second-order quasilinear parabolic equationsyesBelgorod State National Research Universit

Mathematical models of a liquid filtration from reservoirs

This article studies the ltration from reservoirs into porous media under gravity. We start with the exact mathematical model at the microscopic level, describing the joint motion of a liquid in reservoir and the same liquid and the elastic solid skeleton in the porous medium. Then using a homogenization procedure we derive the chain of macroscopic models from the poroelasticity equations up to the simplest Darcy's law in the porous medium and hydraulics in the reservoi

Analysis of models of oil-by-water displacement through the microstructure

Two major arrangements describing the oil-by-water displacement process, i.e., Muskat’s model without taking into account the surface tension on the free boundary and the Buckley-Leverett model based on the surface tension, are considered. These arrangements were subject to theoretical and numerical study, which made it possible to uncover their self-contradictorines

The global-in-time existence of a classical solution for some free boundary problem

The article addresses the issue of whatFree (unknown) boundary problems for differential equations are among the most difficult in the theory of partial differential equation